The problem of internal stabilization for the parabolic system in cascade that we study here becomes more difficult for more than 2 states and distinct diffusion coefficients. The determination of the stabilizing control relies on an existence of a coordinates transformation. In [1]}, a dual internal observer design problem for parabolic systems was studied for the \(3\times 3\)  nonlinear case while in [2]} and [3]}, Chapter 3, the cases of \(3\times 3\)  and \(m\times m\)  linear space-varying hyperbolic systems were studied. In these works, appropriate infinite-dimensional state transformations were introduced to deal with distinct elements on the diagonal of the coefficient of systems' differential operators leading to the use of higher-order spatial derivatives of the measurement.
In the present work, to solve similar problems, we follow a modal decomposition approach. We generalize this method mainly used for the scalar case (see [4]} on direct Lyapunov method for state feedback, see also [5]}) or for the vector case but with controls placed in all states (see [6]} for boundary stabilization), to the case of underactuated systems. We assume that the number of internal inputs is equal to the number of unstable modes and that the resulting matrix that multiplies control inputs in the unstable modes is non-singular. We then introduce a novel state transformation for ODEs with dimension equal to the number of coupled PDEs and written as a polynomial matrix in the slower eigenvalues (needed to be stabilized at given decay rate) of the parabolic operator with order related to the number of distinct diffusion coefficients. The coefficients of this polynomial matrix are nilpotent matrices up to identity matrix, which are easily determined via a provided algorithm, while their values depend on the dynamics of the parabolic system. The stabilizing law simply relies on the determination of stabilizing gains for the zero-order coupling matrix and also on our introduced state transformation depending on the dynamics only.
In this way, any system specification can lead to a construction of unified and scalable control laws independently of the number of eigenvalues needed to be stabilized, which can be arbitrarily large.
